Linear Dynamical Systems Research Articles

A control theoretic approach to evaluate and inform ecological momentary interventions.

Ecological momentary interventions (EMI) are digital mobile health interventions administered in an individual's daily life to improve mental health by tailoring intervention components to person and context. Experience sampling via ecological momentary assessments (EMA) furthermore provides dynamic contextual information on an individual's mental health state. We propose a personalized data-driven generic framework to select and evaluate EMI based on EMA. We analyze EMA/EMI time-series from 10 individuals, published in a previous study. The EMA consist of multivariate psychological Likert scales. The EMI are mental health trainings presented on a smartphone. We model EMA as linear dynamical systems (DS) and EMI as perturbations. Using concepts from network control theory, we propose and evaluate three personalized data-driven intervention delivery strategies. Moreover, we study putative change mechanisms in response to interventions. We identify promising intervention delivery strategies that outperform empirical strategies in simulation. We pinpoint interventions with a high positive impact on the network, at low energetic costs. Although mechanisms differ between individuals - demanding personalized solutions - the proposed strategies are generic and applicable to various real-world settings. Combined with knowledge from mental health experts, DS and control algorithms may provide powerful data-driven and personalized intervention delivery and evaluation strategies.

Fast Iterative Sample Transfer Identification Method for Dynamic Systems Under Non‐identical Distribution

ABSTRACTThis paper proposes a method to improve the identification performance of linear dynamic systems by utilizing knowledge from samples of non‐identical distribution systems. Traditional identification methods heavily rely on the quality of the dataset, such as sample length and noise level, which constrains their performance due to the assumption of identical distribution. Motivated by the concept of sample‐based transfer learning, we propose a sample transfer identification method and derive the condition to avoid negative transfer. We develop a fast iterative transfer identification method for low storage costs, considering the computational burden imposed by the sample size from the source system. Additionally, based on the fast iterative transfer identification method, considering the need to update the current measurement data model in real time, a fast iterative online sample transfer identification method is explored. Through simulations, we validate the effectiveness and superiority of the proposed methods. The results show that sample transfer identification is superior to non‐transfer identification and fast iterative sample transfer identification effectively reduces the calculation amount when dealing with low quality measurement data.

Linear RNNs Provably Learn Linear Dynamical Systems

Linear RNNs Provably Learn Linear Dynamical Systems

Humans actively reconfigure neural task states.

The ability to switch between different tasks is a core component of adaptive cognition, but a mechanistic understanding of this capacity has remained elusive. Longstanding questions over whether task switching requires active preparation remain hotly contested, in large part due to the difficulty of inferring preparatory dynamics from behavior or time-locked neuroimaging. We make progress on this debate by quantifying neural task representations using high-dimensional linear dynamical systems fit to human electroencephalographic recordings. We find that these dynamical systems have high predictive accuracy and reveal neural signatures of active preparation that are shared with task-optimized neural networks. These findings inform a classic debate about how we control our cognition, and offer a promising new paradigm for neuroimaging analysis.

Quantum algorithm for dynamic mode decomposition integrated with a quantum differential equation solver

We present a quantum algorithm that analyzes time series data simulated by a quantum differential equation solver. The proposed algorithm is a quantum version of a dynamic mode decomposition algorithm used in diverse fields such as fluid dynamics, molecular dynamics, and epidemiology. Our quantum algorithm can also compute matrix eigenvalues and eigenvectors by analyzing the corresponding linear dynamical system. Our algorithm handles a broad range of matrices, particularly those with complex eigenvalues. The complexity of our quantum algorithm is O(polylogN) for an N-dimensional system. This is an exponential speedup over known classical algorithms with at least O(N) complexity. Thus, our quantum algorithm is expected to enable high-dimensional dynamical systems analysis and large matrix eigenvalue decomposition, intractable for classical computers. Published by the American Physical Society 2024

Приближенный метод синтеза непрерывных систем совместного оценивания и управления на основе SDRE технологи

<p>The problem of approximate synthesis of a closed loop nonlinear continuous system of joint estimation and control is considered. An approach is used based on the application of the idea of the separation theorem for linear dynamical systems. Using the factorization operation, a nonlinear system is transformed into a structure similar to a linear system, and algorithms for synthesizing an optimal linear controller and state observer are applied to the transformed system, a feature of which is the dependence of the matrices included in the corresponding Riccati equations on the state vector. An example of the synthesis of a state observer and a controller is given, demonstrating the application of the proposed algorithm.</p>

Traffic advisory for ship encounter situation based on linear dynamic system

Enhancing Situation Awareness (SA) is crucial for maritime traffic safety. Various indicators have been developed to assess risks in encounter situations and support the SA of Vessel Traffic Service Operators (VTSOs) and Officers on Watch (OOW), including collision risk and traffic complexity. Despite the widespread use of these navigational aids, ship collision incidents have not been effectively reduced. This paper abstracts ship encounter situations as linear dynamic systems to enhance the understanding of traffic situations. A traffic advisory framework based on the Convention on the International Regulations for Preventing Collisions at Sea (COLREGs) is proposed by integrating complexity metrics with risk indicators. The proposed method is validated through simulations of head-on, overtaking, and crossing scenarios, demonstrating its ability to accurately assess encounter complexity and issue advisories for free navigation, complexity, and resolution. Finally, we discuss the practical application of the proposed method through real-world experiments conducted in the waters of Qiongzhou Strait. The results indicate that the proposed method effectively quantifies the complexity of ship encounter situations and identifies high-collision-risk vessels from a microscopic perspective while providing insights into maritime traffic surveillance from a macro perspective.

Phase trajectories, chaotic behavior, and solitary wave solutions for (3+1)-dimensional integrable Kadomtsev–Petviashvili equation in fluid dynamics

This paper focuses on finding exact soliton solutions and further examining the qualitative characteristics of these solutions for a nonlinear partial differential equation known as the (3+1)-dimensional integrable Kadomtsev–Petviashvili equation that portrays a unique dispersion effect. The (3+1)-dimensional integrable Kadomtsev–Petviashvili equation models the evolution of weakly nonlinear dispersive waves in three spatial dimensions “x,y,z” and time “t”. It balances nonlinearity and dispersion, leading to solitary wave solutions that are stable and localized. This equation is crucial in understanding complex wave phenomena in fields like fluid dynamics, plasma physics, and optics. Initially, the traveling wave transformation is employed to convert the nonlinear partial differential equations into the non-linear ordinary differential equations. Then, we use the Riccati equation approach to find solitary wave solutions for the resulting ordinary differential equation. To better comprehend the physical significance of these derived solutions, we illustrate them using various visual representations. Diverse novel collections of solitary wave solutions are obtained such as bright solitons, dark solitons, and dark singular solitons. Second, we convert the nonlinear ordinary differential equation into a linear dynamical system using the planar dynamical transformation. Then, we undertake a qualitative analysis of the dynamical system to study its chaotic characteristics and bifurcation phenomena. Phase portraits of bifurcation are observed at the equilibrium points of the planner dynamical system. We discuss various tools to identify the chaos (random and unpredictable behavior) in dynamical systems. Graphical visualization of qualitative analysis of the system is also depicted in 3-D phase portraits, 2-D phase portraits, Poincaré mapping, and time series. Additionally, the Runge–Kutta technique is used to do a thorough sensitivity analysis of the dynamical system. Using this analytical procedure, it is verified that the stability of the solution has negligible impact while there is little change in the beginning circumstances. This paper aims to provide vital views to scientists and researchers seeking to advance their experimental operations. Furthermore, this research might be expanded by incorporating solitons and situations and investigating wave dynamics. The methods employed provide a direct and uncomplicated approximation of all solutions when compared to the previously established techniques. Various combinations and values of the physical parameters are utilized to explore the optical soliton solutions of the resultant system.

Quantized State Estimation for Linear Dynamical Systems.

This paper investigates state estimation methods for dynamical systems when model evaluations are performed on resource-constrained embedded systems with finite precision compute elements. Minimum mean square estimation algorithms are reformulated to incorporate finite-precision numerical errors in states, inputs, and measurements. Quantized versions of least squares batch estimation, sequential Kalman, and square-root filtering algorithms are proposed for fixed-point implementations. Numerical simulations are used to demonstrate performance improvements over standard filter formulations. Steady-state covariance analysis is employed to capture the performance trade-offs with numerical precision, providing insights into the best possible filter accuracy achievable for a given numerical representation. A low-latency fixed-point acceleration state estimation architecture for optomechanical sensing applications is realized on Field Programmable Gate Array System on Chip (FPGA-SoC) hardware. The hardware implementation results of the estimator are compared with double-precision MATLAB implementation, and the performance metrics are reported. Simulations and the experimental results underscore the significance of modeling quantization errors into state estimation pipelines for fixed-point embedded implementations.

Dynamic mode decomposition based on expectation–maximization algorithm for simultaneous system identification and denoising

The present study proposes a novel dynamic mode decomposition (DMD) that can simultaneously estimate the reduced-order model, the original signal, and the system/observation noise model only from the noisy data. An expectation–maximization (EM)-algorithm DMD (EMDMD) combines DMD and the parameter adjustment of the linear dynamical system (LDS) based on the EM algorithm. The initial parameters based on the linearity of the reduced-order data are set by using DMD. Subsequently, the log-likelihood of the complete data is maximized by adjusting the LDS parameters while separating the noise. The proposed algorithm is applied to the benchmark data of the short-fat and tall-skinny data matrices with different noise and the time-series velocity fields of the flow around a circular cylinder and the separated flow around an airfoil. The performance of EMDMD in terms of system identification and noise separation from the noisy data is evaluated, and the EMDMD shows the highest system identification and noise separation performance in all data.

Fixed-time active fault-tolerant control for dynamical systems with intermittent faults and unknown disturbances

In this article, the problem of fixed-time active fault-tolerant control is investigated for dynamical linear systems with intermittent faults and unknown disturbances. Unlike traditional active fault-tolerant control, fixed-time control is taken into account in this article since intermittent faults appear and disappear within a certain period of time. The entire active fault-tolerant control framework is composed of fault detection, fault isolation, fault and state estimation as well as the reconfigurable controller. Using the homogeneity-based observers, states and faults are well estimated and a fault diagnosis scheme is proposed for the sake of detecting and isolating intermittent faults in a fixed time. The fault-tolerant controller, which provides global practical fixed-time stability of the closed-loop system, has two switching states corresponding to the appearance and disappearance of intermittent faults. As a consequence, intermittent faults are compensated via the designed active fault-tolerant control method and the system reaches practical stability with the entire convergence time bounded in a fixed time. Finally, two examples are exploited to demonstrate the effectiveness of theoretical results.

Controllability of Volterra integro-dynamic systems with prescribed controls on time scales

The controllability analysis of semilinear Volterra integro-dynamic systems with prescribed controls that are defined on time scales is the focus of this work. We first establish the necessary and sufficient conditions for the controllability of the corresponding linear dynamic system. We next derive sufficient conditions for the controllability of semilinear integro-dynamic systems via Banach fixed point theorem. To demonstrate the practical implications of our findings, we also provide numerical examples for linear and semilinear dynamic systems defined on different time scales.

On hybrid dynamical systems of differential–difference equations

In this paper, we define and study a class of linear hybrid dynamical systems characterized by differential–difference equations. We introduce two operators that facilitate the analysis of these systems and derive explicit formulas for their solutions. We examine the transfer function matrix and characteristic polynomial to assess stability. Our theoretical findings are supported by numerical examples, demonstrating their application in power systems stability analysis. Specifically, we substantiate our theory within the context of power systems stability analysis, incorporating elements of discrete behavior.

Moment-Based Reinforcement Learning for Ensemble Control.

Problems involving controlling the collective behavior of a population of structurally similar dynamical systems, the so-called ensemble control, arise in diverse emerging applications and pose a grand challenge in systems science and control engineering. Owing to the severely under-actuated nature and the difficulty of placing large-scale sensor networks, ensemble systems are limited to being actuated and monitored at the population level. Moreover, mathematical models describing the dynamics of ensemble systems are often elusive. Therefore, it is essential to design broadcast controls that excite the entire population in such a way that the heterogeneity in system dynamics is robustly compensated. In this article, we propose a reinforcement learning (RL)-based data-driven control framework incorporating population-level aggregated measurement data to learn a global control signal for steering a dynamic population in the desired manner. In particular, we introduce the notion of ensemble moments induced by aggregated measurements and derive the associated moment system to the original ensemble system. Then, using the moment system, we learn an approximation of optimal value functions and the associated policies in terms of ensemble moments through RL. We illustrate the feasibility and scalability of the proposed moment-based approach via numerical experiments using a population of linear, bilinear, and nonlinear dynamic ensemble systems. We report that the proposed method achieves the desired control objectives of various ensemble control tasks and obtains significantly better averaged-reward when compared with three existing methods.

On Special Quadratic Lyapunov Functions for Linear Dynamical Systems With an Invariant Cone

On Special Quadratic Lyapunov Functions for Linear Dynamical Systems With an Invariant Cone

Strong versions of impulsive controllability and sampled observability

We give a simple proof of the (perhaps not so) well known fact that exponential polynomials of order k with real exponents have at most k−1 real zeros. We deduce several results that relate to impulsive controllability and sampled observability of finite-dimensional linear time-invariant dynamical systems. We prove that the initial state of a continuous-time linear time-invariant dynamical system of dimension n can be uniquely reconstructed from the sampled output regardless of its sampling time sequence of length n if and only if the system is observable and all the eigenvalues of the system matrix are real. This result thus characterizes sampled observability with arbitrary sampling times. Likewise, we prove that the system is controllable by means of n impulses regardless of when they occur if and only if the system is controllable and all the eigenvalues of the system matrix are real. We also show that, if the system is observable and all the eigenvalues of the system matrix are real, then there exists an initial state such that the times where the output crosses a prescribed threshold are prescribed m times with m<n.

GA based order abatement technique for linear dynamic systems for continuous time system and discrete time system

GA based order abatement technique for linear dynamic systems for continuous time system and discrete time system

Flow control by a hybrid use of machine learning and control theory

PurposeFlow control has a great potential to contribute to a sustainable society through mitigation of environmental burden. However, the high dimensional and nonlinear nature of fluid flows poses challenges in designing efficient control laws using the control theory. This paper aims to propose a hybrid method (i.e. machine learning and control theory) for feedback control of fluid flows, by which the flow is mapped to the latent space in such a way that the linear control theory can be applied therein.Design/methodology/approachThe authors propose a partially nonlinear linear system extraction autoencoder (pn-LEAE), which consists of convolutional neural networks-based autoencoder (CNN-AE) and a custom layer to extract low-dimensional latent dynamics from fluid velocity field data. This pn-LEAE is designed to extract a linear dynamical system so that the modern control theory can easily be applied, while a nonlinear compression is done with the autoencoder (AE) part so that the latent dynamics conform to that linear system. The key technique is to train this pn-LEAE with the ground truths at two consecutive time instants, whereby the AE part retains its capability as the AE, and the weights in the linear dynamical system are trained simultaneously.FindingsThe authors demonstrate the effectiveness of the linear system extracted by the pn-LEAE, as well as the designed control law’s effectiveness for a flow around a circular cylinder at the Reynolds number of ReD = 100. When the control law derived in the latent space was applied to the direct numerical simulation, the lift fluctuations were suppressed over 50%.Originality/valueTo the best of the authors’ knowledge, this is the first attempt using CNN-AE for linearization of fluid flows involving transient development to design a feedback control law.

On Generalized Two Dimensional Linear Hybrid Dynamical Systems

On Generalized Two Dimensional Linear Hybrid Dynamical Systems

Modeling, Analysis, and Control of Autonomous Mobility-on-Demand Systems: A Discrete-Time Linear Dynamical System and a Model Predictive Control Approach

Modeling, Analysis, and Control of Autonomous Mobility-on-Demand Systems: A Discrete-Time Linear Dynamical System and a Model Predictive Control Approach